Optimal. Leaf size=250 \[ -\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}} \]
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Rubi [A]
time = 0.12, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {81, 52, 65, 223,
212} \begin {gather*} \frac {(b d-a e)^3 (5 a B e-8 A b e+3 b B d) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}}-\frac {\sqrt {a+b x} \sqrt {d+e x} (b d-a e)^2 (5 a B e-8 A b e+3 b B d)}{64 b^3 e^2}-\frac {(a+b x)^{3/2} \sqrt {d+e x} (b d-a e) (5 a B e-8 A b e+3 b B d)}{32 b^3 e}-\frac {(a+b x)^{3/2} (d+e x)^{3/2} (5 a B e-8 A b e+3 b B d)}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 212
Rule 223
Rubi steps
\begin {align*} \int \sqrt {a+b x} (A+B x) (d+e x)^{3/2} \, dx &=\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left (4 A b e-B \left (\frac {3 b d}{2}+\frac {5 a e}{2}\right )\right ) \int \sqrt {a+b x} (d+e x)^{3/2} \, dx}{4 b e}\\ &=-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}-\frac {((b d-a e) (3 b B d-8 A b e+5 a B e)) \int \sqrt {a+b x} \sqrt {d+e x} \, dx}{16 b^2 e}\\ &=-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}-\frac {\left ((b d-a e)^2 (3 b B d-8 A b e+5 a B e)\right ) \int \frac {\sqrt {a+b x}}{\sqrt {d+e x}} \, dx}{64 b^3 e}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}} \, dx}{128 b^3 e^2}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {d-\frac {a e}{b}+\frac {e x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{64 b^4 e^2}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {\left ((b d-a e)^3 (3 b B d-8 A b e+5 a B e)\right ) \text {Subst}\left (\int \frac {1}{1-\frac {e x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {d+e x}}\right )}{64 b^4 e^2}\\ &=-\frac {(b d-a e)^2 (3 b B d-8 A b e+5 a B e) \sqrt {a+b x} \sqrt {d+e x}}{64 b^3 e^2}-\frac {(b d-a e) (3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} \sqrt {d+e x}}{32 b^3 e}-\frac {(3 b B d-8 A b e+5 a B e) (a+b x)^{3/2} (d+e x)^{3/2}}{24 b^2 e}+\frac {B (a+b x)^{3/2} (d+e x)^{5/2}}{4 b e}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.75, size = 231, normalized size = 0.92 \begin {gather*} \frac {\sqrt {a+b x} \sqrt {d+e x} \left (15 a^3 B e^3-a^2 b e^2 (31 B d+24 A e+10 B e x)+a b^2 e \left (16 A e (4 d+e x)+B \left (9 d^2+20 d e x+8 e^2 x^2\right )\right )+b^3 \left (8 A e \left (3 d^2+14 d e x+8 e^2 x^2\right )+B \left (-9 d^3+6 d^2 e x+72 d e^2 x^2+48 e^3 x^3\right )\right )\right )}{192 b^3 e^2}+\frac {(b d-a e)^3 (3 b B d-8 A b e+5 a B e) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{64 b^{7/2} e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs.
\(2(212)=424\).
time = 0.09, size = 968, normalized size = 3.87
method | result | size |
default | \(\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (24 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b \,e^{4}-24 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{3} e -62 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} b d \,e^{2}+18 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a \,b^{2} d^{2} e -20 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} b \,e^{3} x +12 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d^{2} e x +224 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d \,e^{2} x +32 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a \,b^{2} e^{3} x +96 B \,b^{3} e^{3} x^{3} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+128 A \,b^{3} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}-15 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{4} e^{4}+9 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{4} d^{4}-48 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{2} b \,e^{3}+48 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d^{2} e +36 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{3} b d \,e^{3}-18 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d^{2} e^{2}-12 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{3} e -72 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} b^{2} d \,e^{3}+72 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a \,b^{3} d^{2} e^{2}+30 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a^{3} e^{3}-18 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} d^{3}+40 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, a \,b^{2} d \,e^{2} x +16 B a \,b^{2} e^{3} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+144 B \,b^{3} d \,e^{2} x^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+128 A a \,b^{2} d \,e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\right )}{384 e^{2} \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{3} \sqrt {b e}}\) | \(968\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.03, size = 734, normalized size = 2.94 \begin {gather*} \left [\frac {{\left (3 \, {\left (3 \, B b^{4} d^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {b} e^{\frac {1}{2}} \log \left (b^{2} d^{2} + 4 \, {\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {x e + d} \sqrt {b} e^{\frac {1}{2}} + {\left (8 \, b^{2} x^{2} + 8 \, a b x + a^{2}\right )} e^{2} + 2 \, {\left (4 \, b^{2} d x + 3 \, a b d\right )} e\right ) - 4 \, {\left (9 \, B b^{4} d^{3} e - {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b - 24 \, A a^{2} b^{2} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} e^{4} - {\left (72 \, B b^{4} d x^{2} + 4 \, {\left (5 \, B a b^{3} + 28 \, A b^{4}\right )} d x - {\left (31 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d\right )} e^{3} - 3 \, {\left (2 \, B b^{4} d^{2} x + {\left (3 \, B a b^{3} + 8 \, A b^{4}\right )} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{768 \, b^{4}}, -\frac {{\left (3 \, {\left (3 \, B b^{4} d^{4} - 4 \, {\left (B a b^{3} + 2 \, A b^{4}\right )} d^{3} e - 6 \, {\left (B a^{2} b^{2} - 4 \, A a b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} d e^{3} - {\left (5 \, B a^{4} - 8 \, A a^{3} b\right )} e^{4}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (b d + {\left (2 \, b x + a\right )} e\right )} \sqrt {b x + a} \sqrt {-b e} \sqrt {x e + d}}{2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} e^{2} + {\left (b^{2} d x + a b d\right )} e\right )}}\right ) + 2 \, {\left (9 \, B b^{4} d^{3} e - {\left (48 \, B b^{4} x^{3} + 15 \, B a^{3} b - 24 \, A a^{2} b^{2} + 8 \, {\left (B a b^{3} + 8 \, A b^{4}\right )} x^{2} - 2 \, {\left (5 \, B a^{2} b^{2} - 8 \, A a b^{3}\right )} x\right )} e^{4} - {\left (72 \, B b^{4} d x^{2} + 4 \, {\left (5 \, B a b^{3} + 28 \, A b^{4}\right )} d x - {\left (31 \, B a^{2} b^{2} - 64 \, A a b^{3}\right )} d\right )} e^{3} - 3 \, {\left (2 \, B b^{4} d^{2} x + {\left (3 \, B a b^{3} + 8 \, A b^{4}\right )} d^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {x e + d}\right )} e^{\left (-3\right )}}{384 \, b^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1386 vs.
\(2 (222) = 444\).
time = 1.46, size = 1386, normalized size = 5.54 \begin {gather*} -\frac {\frac {192 \, {\left (\frac {{\left (b^{2} d - a b e\right )} e^{\left (-\frac {1}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} A a d {\left | b \right |}}{b^{2}} - \frac {8 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} B d {\left | b \right |}}{b} - \frac {8 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} B a {\left | b \right |} e}{b^{2}} - \frac {8 \, {\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )}}{b^{2}} + \frac {{\left (b^{6} d e^{3} - 13 \, a b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{7} d^{2} e^{2} + 2 \, a b^{6} d e^{3} - 11 \, a^{2} b^{5} e^{4}\right )} e^{\left (-4\right )}}{b^{7}}\right )} - \frac {3 \, {\left (b^{3} d^{3} + a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3}\right )} e^{\left (-\frac {5}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {3}{2}}}\right )} A {\left | b \right |} e}{b} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {6 \, {\left (b x + a\right )}}{b^{3}} + \frac {{\left (b^{12} d e^{5} - 25 \, a b^{11} e^{6}\right )} e^{\left (-6\right )}}{b^{14}}\right )} - \frac {{\left (5 \, b^{13} d^{2} e^{4} + 14 \, a b^{12} d e^{5} - 163 \, a^{2} b^{11} e^{6}\right )} e^{\left (-6\right )}}{b^{14}}\right )} + \frac {3 \, {\left (5 \, b^{14} d^{3} e^{3} + 9 \, a b^{13} d^{2} e^{4} + 15 \, a^{2} b^{12} d e^{5} - 93 \, a^{3} b^{11} e^{6}\right )} e^{\left (-6\right )}}{b^{14}}\right )} \sqrt {b x + a} + \frac {3 \, {\left (5 \, b^{4} d^{4} + 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 20 \, a^{3} b d e^{3} - 35 \, a^{4} e^{4}\right )} e^{\left (-\frac {7}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{b^{\frac {5}{2}}}\right )} B {\left | b \right |} e}{b} - \frac {48 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} B a d {\left | b \right |}}{b^{3}} - \frac {48 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} A d {\left | b \right |}}{b^{2}} - \frac {48 \, {\left (\frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} e^{\left (-\frac {3}{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} e^{\frac {1}{2}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b}} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + {\left (b d e - 5 \, a e^{2}\right )} e^{\left (-2\right )} + 2 \, a\right )} \sqrt {b x + a}\right )} A a {\left | b \right |} e}{b^{3}}}{192 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,\sqrt {a+b\,x}\,{\left (d+e\,x\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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